The determinant quantum Monte Carlo method [1]}, [2]}, [3]}, [4]} is a powerful technique to find equilibrium properties of interacting fermions. It is controlled, in the sense that the results are guaranteed to converge to the exact answers upon increasing the computational effort. Unfortunately, in many cases of interest, the applicability of the DQMC method is limited due to the fermion sign problem [5]}. In these cases, the computational cost for a given accuracy increases exponentially with the system size and the inverse temperature. The repulsive Hubbard model suffers from the fermion sign problem in DQMC at generic values of the filling. Below, we briefly review some useful results that were nevertheless obtained using DQMC, either for special parameters where the sign problem is absent, or by employing very large computational resources to overcome the sign problem.
